Distributional SAdS BH spacetime-induced vacuum dominance 3ДОМRED1349769084-SDI Paper Template 2003

Distributional SAdS BH spacetime-induced vacuum dominance

Jaykov Foukzon 1*

[email protected] for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel

Haifa, 3200003

Alexander Potapov 2

[email protected]’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences,

Moscow, RussiaMoscow, 125009

Elena Men’kova 3

[email protected] Research Institute for Optical and Physical Measurements, Moscow, Russia

Moscow, 119361.ABSTRACT

This paper dealing with extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1]-[2], [5]-[7] and [14]-[15]. The regularization of singularities presented in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates. However the Tolman formula for the total energy of a static and asymptotically flat spacetime, gives as it should be. The vacuum energy density of free scalar quantum field with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such distributional BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations has a singular

behavior on BHs horizon :

* Tel.: +xx xx 265xxxxx; fax: +xx aa 462xxxxx.E-mail address: [email protected].

1

1

2

3

45678910111213141516171819202122232425

23

Keywords: Colombeau nonlinear generalized functions, Distributional Riemannian Geometry, Distributional Schwarzschild Geometry, Schwarzschild singularity, Schwarzschild Horizon, smooth regularization, nonsmooth regularization, quantum fields on curved spacetime, vacuum fluctuations, vacuum dominance1. INTRODUCTION 1.1. The breakdown of canonical formalism of Riemann geometry for the

singular solutions of the Einstein field equationsEinstein field equations was originally derived by Einstein in 1915 in respect with canonical formalism of Riemann geometry, i.e. by using the classical sufficiently smooth metric tensor, smooth Riemann curvature tensor, smooth Ricci tensor, smooth scalar curvature, etc. However, singular solutions of the Einstein field equations with singular metric tensor and singular Riemann curvature tensor have soon been found. These singular solutions are formally accepted beyond rigorous canonical formalism of Riemannian geometry.

Remark 1.1. Note that if some components of the Riemann curvature tensor

become infinite at point one obtains the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood of the point i.e. in such

neighborhood Riemann curvature tensor will be changed by formula (1.7) see remark 1.2.

Remark 1.2. Let be infinitesimal closed contour and let be the corresponding surface

spanning by , see figure 1. We assume now that: (i) Christoffel symbol becomes

infinite at singular point by formulae

and (ii)

Let us derive now to similarly canonical calculation [3]-[4] the general formula for

the regularized change in a vector after parallel displacement around

infinitesimal closed contour This regularized change can clearly be written in the form

where

and where the integral is taken over the given

contour . Substituting in place of the canonical expression (see [4], Eq. (85.5)) we obtain

where


4

26272829303132333435363738

3940414243

444546

47

48495051

52

53

5455

56

57

56

Fig. 1. Infinitesimal closed contour and corresponding singular surface spanning by . Due to the degeneracy of the metric (1.12) at point , the Levi-

Civita’ connection is not available on

in canonical sense but only in an distributional sense

Fig. 2. Infinitesimal closed contour with a singularity at point on Horizon and corresponding singular surface spanning by . Due to the degeneracy of the metric (1.12) at , the Levi-Civita’ connection

is not available on horizon in

canonical sense but only in an distributional sense

Now applying Stokes' theorem (see [4], Eq. (6.19)) to the integral (1.3) and considering that the area enclosed by the contour has the infinitesimal value we get


7

585960

61

62

63646566

67

68697071

72

89

Substituting the values of the derivatives (1.4) into Eq. (1.5), we get finally:

where is a tensor of the fourth rank

Here is the classical Riemann curvature tensor. That is a tensor is clear from the

fact that in (1.6) the left side is a vector -- the difference between the values of vectors at one and the same point. Note that a similar results were obtained by many authors [5]-[17] by using Colombeau nonlinear generalized functions [1]-[2].

Definition 1.1. The tensor is called the generalized curvature tensor or the generalized Riemann tensor.

Definition 1.2. The generalized Ricci curvature tensor is defined as

Definition 1.3. The generalized Ricci scalar is defined as

Definition 1.4. The generalized Einstein tensor is defined as

Remark 1.3. (I) Note that the Schwarzschild spacetime is well defined only for . The boundary of the manifold in is the submanifold of


10

73

74

75

76

77

78

79

80818283

84

8586

87

88

89

90

9192

93

94

95

96

9798

1112

diffeomorfic to a product This submanifold is called the event horizon, or simply the horizon [33], [34].

(II) The Schwarzschild metric (1.12) in canonical coordinates with ceases to be a smooth Lorentzian metric for because for such a value of the coefficient becomes zero while becomes infinite. For the metric (1.12) is again a smooth Lorentzian metric but is a space coordinate while is a time coordinate. Hence the metric cannot be said to be either spherically symmetric or static for [33].

(III) From consideration above obviously follows that on Schwarzschild spacetime the Levi-Civita connection

is not available in classical sense and that is well known many years from mathematical literature, see for example [22] section 6 and Remark 1.1 - Remark 1.2 above.

(IV) Note that [4]: (i) The determinant of the metric (1.12) is reqular on horizon, i.e. smooth and non-vanishing for

In addition:

(i) The curvature scalar is zero for

(ii) The none of higher-order scalars such as etc. blows up. For example

the quadratic scalar is regular on horizon, i.e. smooth and non-vanishing for

(V) Note that: (i) In physical literature (see for example [4], [33], [35],) it was wrongly assumed that a properties (i)-(iii) is enough to convince us that represent an non honest physical singularity but only coordinate singularity.

(VI) Such assumption based only on formal extensions

of the curvature scalar and higher-order scalars such as

on horizon and on origin by formulae


13

99100

101

102103104105106107108

109

110

111

112113

114115116

117

118

119

120121122123124125

126127128129

130

1415

however in the limit the Levi-Civitá connection becomes infinite [4]:

Thus obviously by consideration above (see Remark 1.1 - Remark 1.2) this extension given by Eq. (1.15) has no any sense in respect of the canonical Riemannian geometry.

(VII) From consideration above (see Remark 1.1 - Remark 1.2) obviously follows that the scalars such as has no any sense in respect of the canonical Levi-Civitá connection (1.11) and therefore cannot be said to be either honest physical singularity or only coordinate singularity in respect of the canonical Riemannian geometry.

Remark 1.4. Note that in physical literature the spacetime singularity usually is defined as location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the classical scalar invariant curvatures of singular spacetime, which includes a measure of the density of matter.

Remark 1.5. In general relativity, many investigations have been derived with regard to singular exact vacuum solutions of the Einstein equation and the singularity structure of space-time. Such solutions have been formally derived under condition where

represent the energy-momentum densities of the gravity source. This for example is the case for the well-known Schwarzschild solution, which is given by, in the Schwarzschild coordinates


16

131

132

133

134

135136

137138139140141

142143144145146

147148149150151152

1718

where is the Schwarzschild radius with and being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. The metric (1.12) describes the gravitational field produced by a point-like particle located at .

Remark 1.6. Note that when we say, on the basis of the canonical expression of the curvature square

formally obtained from the metric (1.12), that is a singularity of the Schwarzschild space-time, the source is considered to be point-like and this metric is regarded as meaningful everywhere in space-time.

Remark 1.7. From the metric (1.12), the calculation of the canonical Einstein tensor proceeds in a straightforward manner gives for

where Using Eq. (1.14) one formally obtains a boundary conditions

However as pointed out above the canonical expression of the Einstein tensor in a sufficiently small neighborhood of the point and must be replaced by the generalized Einstein tensor (1.10). By simple calculation easy to see that

and therefore the boundary conditions (1.15) is completely wrong. But other hand as pointed out by many authors [5]-[17] that the canonical representation of the Einstein tensor, valid only in a weak (distributional) sense, i.e. [12]:


19

153

154

155156157158159160

161

162

163164165

166167168

169

170

171

172

173174175

176

177178179

180

181

2021

and therefore again we obtain Thus canonical definition of the Einstein tensor is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin .

1.2. The distributional Schwarzschild geometry

General relativity as a physical theory is governed by particular physical equations; the focus of interest is the breakdown of physics which need not coincide with the breakdown of geometry. It has been suggested to describe singularity at the origin as internal point of the Schwarzschild spacetime, where the Einstein field equations are satisfied in a weak (distributional) sense [5]-[22].

1.2.1.The smooth regularization of the singularity at the origin

The two singular functions we will work with throughout this paper (namely the singular

components of the Schwarzschild metric) are and Since it

obviously gives the regular distribution . By convolution with a mollifier

adapted to the symmetry of the spacetime, (i.e. chosen radially symmetric) we embed it into the Colombeau algebra [22]:

Inserting (1.18) into (1.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:

Remark 1.8. Note that under regularization (1.18) for any the metric

obviously is a classical Riemannian object and there is no existed the breakdown of canonical formalism of Riemannian geometry for these metrics, even at origin It has been suggested by many authors to describe singularity at the origin as an internal point,


22

182183184185

186187188189190191192

193

194195196

197

198

199200

201

202

203204

205

206

207

208209

210

211

212213214

2324

where the Einstein field equations are satisfied in a distributional sense [5]-[22]. From the Colombeau metric (1.19) one obtains in a distributional sense [22]:

Hence, the distributional Ricci tensor and the distributional curvature scalar are of δ

-type, i.e.

Remark 1.9. Note that the formulae (1.20) should be contrasted with what is the expected result given by Eq. (1.17). However the equations (1.20) are obviously given in spherical coordinates and therefore strictly speaking this is not correct,

because the basis fields are not globally defined. Representing

distributions concentrated at the origin requires a basis regular at the origin. Transforming the formulae for into Cartesian coordinates associated with the spherical ones, i.e.,

, we obtain, e.g., for the Einstein tensor the expected result

given by Eq. (1.17), see [22].

1.2.2. The nonsmooth regularization of the singularity at the origin

The nonsmooth regularization of the Schwarzschild singularity at the origin is considered by N. R. Pantoja and H. Rago in paper [12]. Pantoja non smooth regularization of the Schwarzschild singularity is

Here is the Heaviside function and the limit is understood in a distributional sense. Equation (1.19) with as given in (1.21) can be considered as a regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.21), the calculation of the distributional Einstein tensor proceeds in a straightforward manner. By simple calculation it gives [12]:


25

215216

217

218

219

220

221222223

224

225226227228

229

230

231232233

234

235

236237238239240

241

2627

and

which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function . Transforming now the formulae for into Cartesian coordinates

associated with the spherical ones, i.e., , we obtain for the generalized Einstein tensor the expected result given by Eq. (1.17)

see Remark 1.9.

1.2.3. The smooth regularization via Horizon

The smooth regularization via Horizon is considered by J. M. Heinzle and R. Steinbauer in

paper [22]. Note that A canonical regularization is the principal value

which can be embedded into [22]:

Inserting now (1.25) into (1.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:


28

242

243

244

245

246247248249

250

251

252

253

254255256

257

258

259

260

261

262263

264

2930

Remark 1.10. Note that obviously Colombeau object, (1.26) is degenerate at because is zero at the horizon. However, this does not come as a surprise. Both and are positive outside of the black hole and negative in the interior. As a consequence any smooth regularization of (or ) must pass through zero somewhere and, additionally, this zero must converge to as the regularization parameter goes to zero.

Remark 1.11. Note that due to the degeneracy of Colombeau object (1.26), even the distributional Levi-Civitá connection obviously is not available by using the smooth regularization via horizon [22].

1.2.4.The nonsmooth regularization via Gorizon

In this paper we leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. The question we are aiming at is the following: using distributional geometry (thus without leaving Schwarzschild coordinates), is it possible to show that the horizon singularity of the Schwarzschild metric is not merely a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at the horizon in Schwarzschild coordinates.

The main focus of this work is a (nonlinear) superdistributional description of the Schwarzschild spacetime. Although the nature of the Schwarzschild singularity is much worse than the quasi-regular conical singularity, there are several distributional treatments in the literature [8]-[29], mainly motivated by the following considerations: the physical interpretation of the Schwarzschild metric is clear as long as we consider it merely as an exterior (vacuum) solution of an extended (sufficiently large) massive spherically symmetric body. Together with the interior solution it describes the entire spacetime. The concept of point particles -- well understood in the context of linear field theories -- suggests a mathematical idealization of the underlying physics: one would like to view the Schwarzschild solution as defined on the entire spacetime and regard it as generated by a point mass located at the origin and acting as the gravitational source.

This of course amounts to the question of whether one can reasonably ascribe distributional curvature quantities to the Schwarzschild singularity at the horizon.

The emphasis of the present work lies on mathematical rigor. We derive the physically expected result for the distributional energy momentum tensor of the Schwarzschild geometry, i.e., , in a conceptually satisfactory way. Additionally, we set up a unified language to comment on the respective merits of some of the approaches taken so far. In particular, we discuss questions of differentiable structure as well as smoothness and degeneracy problems of the regularized metrics, and present possible refinements and workarounds. These aims are accomplished using the framework of nonlinear supergeneralized functions (supergeneralized Colombeau algebras ). Examining


31

265

266267268269270271

272273274

275

276

277

278279280281282283

284285286287288289290291292293294

295296

297298299300301302303304

3233

the Schwarzschild metric (1.12) in a neighborhood of the horizon, we see that, whereas is smooth, is not even (note that the origin is now always excluded from

our considerations; the space we are working on is ). Thus, regularizing the Schwarzschild metric amounts to embedding into (as done in (3.2)). Obviously, (3.1) is degenerated at , because is zero at the horizon. However, this does not come as a surprise. Both and are positive outside of the black

hole and negative in the interior. As a consequence any (smooth) regularization (

) [above (below) horizon] of must pass through small enough vicinity

( )

of zeros set somewhere and, additionally, this vicinity

( ) must converge to as the regularization parameter goes to zero. Due to the degeneracy of (1.12), the Levi-Cività connection is not available. By appropriate nonsmooth regularization (see section 3) we obtain an Colombeau generalized object modeling the singular Schwarzschild metric above and below horizon, i.e.,

Consider corresponding distributional connections

and


34

305306307308309310311312

313

314

315

316

317318319320321

322

323

324

325

326

327

328

329

330

331

332

3536

Obviously coincides with the corresponding Levi-Cività connection

on as and

there. Clearly, connections in respect the regularized metric

i.e. . Proceeding in this manner, we obtain the nonstandard result

Investigating the weak limit of the angular components of the generalized Ricci tensor using the abbreviation

and let be the function ( ) where by (

) we denote the class of all functions with compact support such that

(i) supp (supp )

(ii) Then for any function we get:


37

333

334

335336337338

339

340

341342

343

344

345346

347

348

349

350

351

352

3839

i.e., the Schwarzschild spacetime is weakly Ricci-nonflat (the origin was excluded from our considerations). Furthermore, the Tolman formula [3], [4] for the total energy of a static and asymptotically flat spacetime

with the determinant of the four dimensional metric and the coordinate volume element, gives

as it should be.

The paper is organized in the following way: in section II we discuss the conceptual as well as the mathematical prerequisites. In particular we comment on geometrical matters (differentiable structure, coordinate invariance) and recall the basic facts of nonlinear superdistributional geometry in the context of algebras of supergeneralized functions. Moreover, we derive sensible nonsmooth regularizations of the singular functions to be used throughout the paper. Section III is devoted to these approach to the problem. We present a new conceptually satisfactory method to derive the main result. In this final section III we investigate the horizon and describe its distributional curvature. Using nonlinear superdistributional geometry and supergeneralized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.

1.2.5. Distributional Eddington-Finkelstein spacetime

In physical literature for many years the belief that Schwarzschild spacetime is extendible exists, in the sense that it can be immersed in a larger

spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with In physical literature [4], [33], [34], [35] one considers the formal change of

coordinates obtained by replacing the canonical Schwarzschild time by "retarded time" above horizon given when by

From (1.31) it follows for

The Schwarzschild metric (1.12) above horizon (see section 3) in this coordinate obviously takes the form


40

353354355

356357

358

359

360

361362363364365366367368369370371

372

373374

375376377378379380

381

382

383384

385

386387

388

4142

When we replace (1.33) below horizon by

From (1.36) it follows for

The Schwarzschild metric (1.12) below horizon (see section 3) in this coordinate obviously takes the form

Remark 1.12. (i) Note that the metric (1.33) is defined on the manifold and obviously it is regular Lorentzian metric: its coefficients are smooth.

(ii) The term ensures its non-degeneracy for

(iii) Due to the nondegeneracy of the metric (1.32) the Levi-Civita connection

obviously now available and therefore nonsingular on horizon in contrast Schwarzschild metric (1.12) one obtains [36]:

(iv) In physical literature [4], [33], [34], [35] by using properties (i)-(iii) this spacetime wrongly convicted as a rigorous mathematical extension of the Schwarzschild spacetime.

Remark 1.13. Let us consider now the coordinates: (i) and

(ii)


43

389

390391

392

393394

395

396397

398

399

400401

402403

404

405

406407

408

409

410411

412

413

4445

Obviously both transformations given by Eq. (1.33) and Eq. (1.36) are singular because the both Jacobian of these transformations is singular at

and

Remark 1.14. Note first (i) such singular transformations not allowed in conventional Lorentzian geometry and second

(ii) both Eddington-Finkelstein metrics given by Eq. (1.35) and by Eq. (1.38) again not well defined in any rigorous mathematical sence at

Remark 1.15. From consideration above follows that Schwarzschild spacetime is not extendible, in the sense that it can be immersed in a larger

spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with Thus Eddington-Finkelstein spacetime cannot be considered as an extension of the

Schwarzschild spacetime in any rigorous mathematical sense in respect conventional Lorentzian geometry. Such "extension" is the extension by abnormal definition and nothing more.

Remark 1.16. From consideration above it follows that it is necessary a regularization of the Eq. (1.34) and Eq. (1.37) on horizon. However, obviously only nonsmooth regularization via horizon is possible. Under nonsmooth regularization (see section 3) Eq. (1.34) and Eq. (1.37) takes the form

and


46

414415416

417

418

419

420

421422

423424425426427428429430431

432433434435

436

437

438

439

440

4748

correspondingly. Therefore Eq. (1.41) - Eq. (1.42) take the form

and

From Eq. (1.43) - Eq. (1.44) one obtains generalized Eddington-Finkelstein transformations such as

and

Therefore, Eq. (1.45) - Eq. (1.46) take the form

and


49

441

442

443

444

445

446

447448

449

450

451

452

453

454

455

456

457

458

5051

At point one obtains

and

where Thus generalized Eddington-Finkelstein transformations (1.47) - (1.48) are well defined in sense of Colombeau generalized functions. Therefore Colombeau generalized object modeling the classical Eddington-Finkelstein metric (1.35) – (1.36) above and below gorizon takes the form

It is easily to verify by using formula A.2 (see appendix) that the distributional curvature scalar is singular at as in the case of the distributional Schwarzschild spacetime given by Eq. (1.28). However this is not surprising because the classical Eddington-Finkelstein spacetime and generalized Eddington-Finkelstein specetime given by Eq. (1.53) are essentially different geometrical objects.

2. GENERALIZED COLOMBEAU CALCULUS

2.1 Notation and basic notions from standard Colombeau theory

We use [1], [2], [7] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper will denote an open subset of . Standard Colombeau generalized functions on are defined as equivalence classes of nets of smooth functions


52

459

460461

462

463

464

465

466467468469470

471

472473474475476477478479480481482483484485

5354

(regularizations) subjected to asymptotic norm conditions with respect to for their derivatives on compact sets.

The basic idea of classical Colombeau's theory of nonlinear generalized functions [1], [2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter . Let with for all

where is a separable, smooth orientable Hausdorff manifold of dimension .

Definition 2.1. The classical Colombeau's algebra of generalized functions on is defined as the quotient:

of the space of sequences of moderate growth modulo the space of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where denoting the space of smooth vector fields on ):

Remark 2.1. In the definition the Landau symbol appears, having the following

meaning:

Definition 2.2. Elements of G are denoted by:

Remark 2.2. With componentwise operations ( ) is a fine sheaf of differential algebras with respect to the Lie derivative defined by .

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., iff for all charts ,


55

486487488489490491

492

493494

495

496

497498499500

501

502

503

504

505506

507508

509

510511

512513

5657

where on the open set in the respective estimates Lie derivatives are replaced by partial derivatives.



Remark 2.3. Smooth functions are embedded into simply by the constant embedding , i.e., , hence is a faithful subalgebra of

.

Point Values of a Generalized Functions on . Generalized Numbers

Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field or . Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these numbers.

Definition 2.3. Inserting into yields a well defined element of the ring of constants (also called generalized numbers) (corresponding to resp. ), defined as the set of moderate nets of numbers ( with for some )

modulo negligible nets ( for each ); componentwise insertion of points of

into elements of yields well-defined generalized numbers, i.e., elements of the ring of constants:

(with = or = for = or = ), where


58

514515

516517518519

520

521522523524

525526527528529530531532

533

534535536537538539

540

541

542

543

544

5960

Generalized functions on are characterized by their generalized point values, i.e., by their values on points in the space of equivalence classes of compactly supported nets

with respect to the relation for all ,

where denotes the distance on induced by any Riemannian metric.

Definition 2.4. For and the point value of at the point is

defined as the class of in

Definition 2.5. We say that an element is strictly nonzero if there exists a representative and a such that for sufficiently small. If is strictly

nonzero, then it is also invertible with the inverse The converse is true as well.

Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.

Definition 2.6. Let be an open subset of . On a set

we introduce an equivalence relation:

and denote by the set of generalized points. The set of points with compact support is

Definition 2.7. A generalized function is called associated to zero, on in L. Schwartz sense if one (hence any) representative converges to zero

weakly, i.e.

We shall often write:


61

545546547548

549

550551

552553554

555556557

558

559

560

561

562

563

564

565566

567

568

569570571

572

573

574

575

6263

The -module of generalized sections in vector bundles-especially the space of generalized tensor fields is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

where denotes the space of smooth tensor fields and the tensor product is taken

over the module . Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [1]-[2], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as -multilinear maps taking generalized vector and

covector fields to generalized functions, i.e., as -modules we have

In particular a generalized metric is defined to be a symmetric, generalized -tensor

field (with its index independent of and) whose determinant is

invertible in . The latter condition is equivalent to the following notion called strictly

nonzero on compact sets: for any representative of we have

for all small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: is a generalized metric iff (on every relatively

compact open subset of ) there exists a representative of such that for

fixed (small enough) (resp. ) is a classical pseudo-Riemannian

metric and is invertible in the algebra of generalized functions. A generalized

metric induces a -linear isomorphism from to and the inverse metric

is a well defined element of (i.e., independent of the

representative ). Also the generalized Levi-Civita connection as well as the generalized Riemann-, Ricci- and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives.

2.2 Generalized Colombeau Calculus

We briefly recall the basic generalized Colombeau construction. Colombeau supergeneralized functions on where are defined as equivalence


64

576

577578579580581

582

583

584585586587588589

590591

592

593594595596597598599600601602603604605606607

608

609610611612

6566

classes of nets of smooth functions where (regularizations) subjected to asymptotic norm conditions with respect to for their derivatives on compact sets.

The basic idea of generalized Colombeau's theory of nonlinear supergeneralized functions [1], [2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter . Let with such that:

(i) and

(ii) for all where is a separable, smooth orientable Hausdorff manifold of dimension .

Definition 2.8. The supergeneralized Colombeau's algebra of supergeneralized functions on where is defined as the quotient:

of the space of sequences of moderate growth modulo the space of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where is denoting the space of smooth vector fields on ):

where is denoting the weak Lie derivative in L. Schwartz sense. In the definition the

Landau symbol appears, having the following meaning:

Definition 2.9. Elements of are denoted by:

Remark 2.4. With componentwise operations ( ) is a fine sheaf of differential algebras with respect to the Lie derivative defined by .


67

613614615

616617618

619

620621

622623624

625

626627628629

630

631

632633634

635636

637

638639

6869





Remark 2.5. Smooth functions are embedded into simply by the constant embedding , i.e., , hence is a faithful subalgebra of

.

Point Values of a Supergeneralized Functions on . Supergeneralized Numbers

Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field or . Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these numbers.

Definition 2.10. Inserting into yields a well defined element of the ring of constants (also called generalized numbers) (corresponding to resp. ), defined as the set of moderate nets of numbers ( with for

some ) modulo negligible nets ( for each ); componentwise insertion of

points of into elements of yields well-defined generalized numbers, i.e., elements of the ring of constants:

(with = or = for = or = ), where

Supergeneralized functions on are characterized by their generalized point values, i.e., by their values on points in the space of equivalence classes of compactly supported


70

640641642643

644645646647

648649650651

652653654655656657658659

660

661662663664665666

667

668

669

670

671672

7172

nets with respect to the relation for all

, where denotes the distance on induced by any Riemannian metric.

Definition 2.11. For and the point value of at the point

is defined as the class of in

Definition 2.12. We say that an element is strictly nonzero if there exists a representative and a such that for sufficiently small. If is strictly

nonzero, then it is also invertible with the inverse The converse is true as well.

Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.

Definition 2.13. Let be an open subset of . On a set

we introduce an equivalence relation:

and denote by the set of supergeneralized points. The set of points with compact support is

Definition 2.14. A supergeneralized function is called associated to zero, on in L. Schwartz's sense if one (hence any) representative converges

to zero weakly, i.e.

We shall often write:


73

673674

675676

677678679

680681682

683

684

685

686

687

688

689690

691

692

693694695

696

697

698

699

7475

Definition 2.15. The -module of supergeneralized sections in vector bundles - especially the space of generalized tensor fields - is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

where denotes the space of smooth tensor fields and the tensor product is taken

over the module . Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as -multilinear maps taking generalized vector and

covector fields to generalized functions, i.e., as -modules we have

In particular a supergeneralized metric is defined to be a symmetric, supergeneralized -tensor field (with its index independent of and) whose

determinant is invertible in . The latter condition is equivalent to the

following notion called strictly nonzero on compact sets: for any representative

of we have for all small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: is a generalized metric iff (on every relatively compact open subset of ) there exists a representative

of such that for fixed (small enough) (resp. ) is a

classical pseudo-Riemannian metric and is invertible in the algebra of generalized

functions. A generalized metric induces a -linear isomorphism from

to and the inverse metric is a well defined element of

(i.e., independent of the representative ). Also the supergeneralized Levi-Civita connection as well as the supergeneralized Riemann, Ricci and Einstein tensor of a supergeneralized metric are defined simply by the usual coordinate formulae on the level of representatives.

2.3 Distributional general relativity

We briefly summarize the basics of distributional general relativity, as a preliminary to latter discussion. In the classical theory of gravitation one is led to consider the Einstein field equations which are, in general, quasilinear partial differential equations involving second order derivatives for the metric tensor. Hence, continuity of the first fundamental form is


76

700

701702703704705

706

707

708709710711712713

714715716717718719720721722723724725726727728729730731732733734735736737738739

7778

expected and at most, discontinuities in the second fundamental form, the coordinate independent statements appropriate to consider 3-surfaces of discontinuity in the spacetime manifolfd of General Relativity.

In standard general relativity, the space-time is assumed to be a four-dimensional differentiable manifold endowed with the Lorentzian metric

.

At each point of space-time , the metric can be diagonalized as with , by choosing the coordinate system

appropriately.

In superdistributional general relativity the space-time is assumed to be a four-dimensional differentiable manifold where endowed with the Lorentzian supergeneralized metric

At each point , the metric can be diagonalized as

by choosing the generalized coordinate system appropriately.

The classical smooth curvature tensor is given by

with being the smooth Christoffel symbol. The supergeneralized nonsmooth curvature

tensor is given by

with being the supergeneralized Christoffel symbol. The fundamental classical

action integral is


79

740741742

743744745

746747748

749750751

752

753

754755

756

757

758

759

760

761762

763

764

765766767

768

8081

where is the Lagrangian density of a gravitational source and is the gravitational Lagrangian density given by

Here is the Einstein gravitational constant and is defined by

with . There exists the relation

with

Thus the supergeneralized fundamental action integral is

where is the supergeneralized Lagrangian density of a gravitational source and

is the supergeneralized gravitational Lagrangian density given by

Here is the Einstein gravitational constant and is defined by

with . There exists the relation

with


82

769770

771

772

773

774

775

776

777

778

779

780

781

782

783

784

785786

787

788

789

790

791

792

793

794

795

796

8384

Also, we have defined the classical scalar curvature by

with the smooth Ricci tensor

From the action the classical Einstein equation

follows, where is defined by

with

being the energy-momentum density of the classical gravity source. Thus we have defined the supergeneralized scalar curvature by

with the

supergeneralized Ricci tensor

From the action

the generalized Einstein equation

follows, where

is defined by

with


85

797

798

799

800

801

802

803

804805

806

807

808

809

810

811812

813

814815

816

817

818

819

820

821

8687

being the

supergeneralized energy-momentum density of the supergeneralized gravity source. The classical energy-momentum pseudo-tensor density of the gravitational field is defined by

with

. The supergeneralized energy-momentum pseudo-tensor density of the gravitational field is defined by

with

.

3. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH REGULARIZATION VIA HORIZON

3.1 Calculation of the stress-tensor by using nonsmooth regularization via Horizon

In this section we leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. The question we are aiming at is the following: using distributional geometry (thus without leaving Schwarzschild coordinates), is it possible to show that the horizon singularity of the Schwarzschild metric is not merely only a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at horizon in

Schwarzschild coordinates. In the usual Schwarzschild coordinates the Schwarzschild metric (1.12) takes the form above horizon and below horizon

correspondingly


88

822

823824

825

826827

828

829830831832833834835836837838839840841

842843844845

846

847

848

8990

Remark 3.1. Following the above discussion we consider the metric coefficients and as an element of and embed it into

by replacements above horizon and below horizon correspondingly

Remark 3.2. Note that, accordingly, we have fixed the differentiable structure of the manifold: the usual Schwarzschild coordinates and the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1) are extended on through the horizon. Therefore we have above horizon and below horizon correspondingly

Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) gorizon, i.e.


91

849850851852

853

854

855

856857858859860

861

862

863

864865

9293

The

generalized Ricci tensor above horizon may now be calculated componentwise using the classical formulae

From (3.2) we

obtain


94

866

867868

869

870

871

9596

h r r 2m

r r 2m2 2 1/2

r 2m2 2 1/2

r2,

rh 1 h

r r 2mr r 2m2 2 1/2

r 2m2 2 1/2

r2 1

r 2m2 2r

r 2mr 2m2 2 1/2

r 2m2 2 1/2

r 1 r 2m2 2

r

r 2mr 2m2 2 1/2

1.

hr r 2m

r r 2m2 2 1/2

r 2m2 2 1/2

r2

1r r 2m2 2 1/2

r 2m2

r r 2m2 2 3/2 r 2mr2 r 2m2 2 1/2

r 2mr2 r 2m2 2 1/2

2 r 2m2 2 1/2

r3.

r2h 2rh

r2 1r r 2m2 2 1/2

r 2m2

r r 2m2 2 3/2 r 2mr2 r 2m2 2 1/2

r 2mr2 r 2m2 2 1/2

2 r 2m2 2 1/2

r3

2r r 2mr r 2m2 2 1/2

r 2m2 2 1/2

r2

rr 2m2 2 1/2

rr 2m2

r 2m2 2 3/2 r 2m

r 2m2 2 1/2

r 2mr 2m2 2 1/2

2 r 2m2 2 1/2

r

2r 2m

r 2m2 2 1/22 r 2m2 2 1/2

r

rr 2m2 2 1/2

rr 2m2

r 2m2 2 3/2.

3.5


97

872873

874

9899

Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation

and

let be the function where by we denote the class of all

functions with compact support such that:

(i) supp

(ii) (ii)

Then for any function we get:

By

replacement from (3.6) we obtain

By replacement from (3.7) we obtain the expression

From Eq. (3.8) we obtain

where we have expressed the function as


100

875876

877

878879

880

881

882

883

884

885

886

887

888

889

890

891

892

101102

with . Equations (3.9) - (3.10) give

Thus in

where from Eq. (3.11) we obtain

For we get:

By replacement from (3.13) we obtain



103

893

894

895

896897

898

899

900

901

902

903

904

905

906

907

908

909

104105

From Eq. (3.15)

we obtain


with . Equation (3.17) gives

where use is made of the relation

Thus in we obtain


106

910

911

912

913

914

915

916

917

918919

920

921

922

923

924

107108

The supergeneralized Ricci tensor below horizon may now be calculated

componentwise using the classical formulae

From Eq. (3.21) we obtain

Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation

where is a function with compact support such that

we get:


109

925926

927

928

929

930

931

932933

934

935

936

937938

939

110111

By replacement from Eq. (3.23) we obtain


which is calculated to give




112

940

941942

943

944

945

946

947

948

949

950

951

952

953

954

955

113114

Thus in where from Eq. (3.28) we obtain

For we get:




115

956

957958

959

960

961

962

963

964

965

966

967

968

116117

which is calculated to give



where use is made of the relation

Thus in we obtain


118

969

970

971

972

973

974

975

976

977

978979

980

981

982

983

119120

Using Eqs. (3.12), (3.20), (3.29), (3.37) we obtain

Thus the Tolman formula [3], [4] for the total energy of a static and asymptotically flat spacetime with the determinant of the four dimensional metric and the coordinate volume element, gives

We rewrite now the Schwarzschild metric (3.3) in the form

Using Eq. (A.5) from Eq. (3.40) one obtains for

and


121

984

985

986

987

988

989990991

992

993

994

995

996

997998

999

1000

1001

122123

3.2. Examples of distributional geometries. Calculation of the distributional quadratic scalars by using nonsmooth regularization via Horizon

Let us consider again the Schwarzschild metric (3.1)

We revrite now the Schwarzschild metric (3.43) above Horizon ( ) in the form

Following the above discussion we consider the singular metric coefficient as an element of and embed it into by replacement

Thus above Horizon ( ) the corresponding distributional metric takes the form

We rewrite now the Schwarzschild metric (3.43) below Horizon ( ) in the form


124

1002

1003100410051006

1007

1008

10091010

1011

10121013

1014

1015

1016

1017

1018

125126

Following the above discussion we consider the singular metric coefficient as an element of and embed it into by replacement

Thus below Horizon ( ) the corresponding distributional metric takes the form

From Eq. (3.46) one obtains

From Eq. (3.46) using Eq. (A.5) one obtains

From Eq. (3.51) for one obtains


127

1019

1020

10211022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

10341035

128129

Remark 3.3. Note that curvature scalar again is nonzero but nonsingular.

Let us introduce now the general metric which has the form [11]:

where

Remark 3.4. Note that the coordinates and are time and space coordinates, respectively, only if

In the Cartesian coordinate system with

x 1 rcos sin ,x 2 rsin sin ,x 3 rcos , 3.56

the metric (3.53)-(3.55) takes the form

with given by


130

1036

1037

1038

1039

1040

1041

1042

1043

10441045

1046

1047

1048

1049

10501051

1052

1053

1054

1055

1056

131132


Regularizing the function above gorizon (under condition ) such as

with from Eq. (3.59)-Eq. (3.60) one obtains

Regularizing the function below gorizon (under condition ) such as

with

from Eq. (3.59), Eq. (3.62) one obtains


133

1057

1058

1059

1060

10611062

1063

1064

10651066

1067

10681069

1070

1071

10721073

134135

Remark 3.5. Finally the metric (3.57) becomes the Colombeau object of the form

with given by

Using

now Eq. A2 one obtains that the Colombeau curvature scalars in terms of

Colombeau generalized functions is expressed as

Remark 3.6. Note that (i) on horizon Colombeau scalars well defined and becomes to infinite large Colombeau generalized numbers


136

1074

1075

1076

10771078

1079

10801081

1082

1083

10841085

137138

(ii) for Colombeau scalars well defined and becomes to infinite small

Colombeau generalized numbers

Using now Eq. A2 one obtains that the Colombeau scalars in terms of


Remark 3.7. Note that (i) on horizon Colombeau scalars well defined and becomes to infinite large Colombeau generalized numbers,

(ii) for Colombeau scalars well defined and becomes to infinite small Colombeau generalized numbers.

Using now Eq. A2 one obtains that the Colombeau scalars in terms of



139

1086

10871088

1089

1090

1091

10921093

10941095

1096

1097

140141

Remark 3.8. Note that (i) on horizon Colombeau scalars well defined and becomes to infinite large Colombeau generalized numbers,

(ii) for Colombeau scalars finite

and tends to zero in the limit

Remark 3.9. Note that under generalized transformations such as

and

the metric given by Eq. (3.61) - Eq. (3.64) becomes to Colombeau metric of the form

4. QUANTUM SCALAR FIELD IN CURVED DISTRIBUTIONAL SPACETIME

4.1 Canonical quantization in curved distributional spacetime

Much of formalism can be explained with Colombeau generalized scalar field. The basic concepts and methods extend straightforwardly to distributional tensor and distributional spinor fields. To being with let's take a spacetime of arbitrary dimension with a metric


142

1098

10991100

11011102

1103

11041105

1106

1107

1108

1109

1110

1111

1112

1113

1114

111511161117111811191120

143144

of signature The action for the Colombeau generalized scalar field

is

The corresponding equation of motion is

Here

With explicit, the mass should be replaced by Separating out a time coordinate we can write the action as

The canonical momentum at a time is given by

where labels a point on a surface of constant x , the x argument of is suppressed,

is the unit normal to the surface, and is the determinant of the induced spatial

metric . To quantize, the Colombeau generalized field and its conjugate

momentum are now promoted to hermitian operators and required to satisfy the canonical commutation relation,

Here for any scalar function without the use of a metric volume element. We form now a conserved bracket from two complex Colombeau solutions to the scalar wave equation (4.2) by

where


145

11211122

1123

11241125

1126

1127

1128

1129

1130

11311132

1133

1134

1135

1136

1137

11381139114011411142

1143

114411451146

1147

11481149

1150

146147

This bracket is called the generalized Klein-Gordon inner product, and the

generalized Klein Gordon norm of The generalized current density is

divergenceless, i.e. when the Colombeau generalized functions

and satisfy the KG equation (4.2), hence the value of the integral in (4.7) is

independent of the spacelike surface Σ over which it is evaluated, provided the functions vanish at spatial infinity. The generalized KG inner product satisfies the relations

We define now the annihilation operator associated with a complex Colombeau solution by the bracket of with the generalized field operator

It follows from the hermiticity of that the hermitian conjugate of is given by

From Eq. (4.5) and CCR (4.6) one obtains

Note that from Eq. (4.11) follows

Note that if is a positive norm solution with unit norm hf with, then and

satisfy the commutation relation Suppose now that is a

normalized quantum state satisfying then for each the state

is a normalized eigenstate of the number operator

with eigenvalue The span of all these states defines a Fock

space of the distributional - wavepacket -particle excitations above the state If we want to construct the full Hilbert space of the field theory in curved distributional spacetime, how can we proceed? We should find a decomposition of the space of complex Colombeau solutions to the wave equation (4.2) into a direct sum of a positive norm subspace and its complex conjugate such that all brackets between solutions from the two subspaces vanish. That is, we must find a direct sum decomposition:


148

11511152

11531154

11551156

1157

1158

11591160

1161

1162

1163

1164

1165

1166

1167

1168

1169

1170

1171

1172

1173

1174

1175

1176

117711781179118011811182

149150

such that

and

The condition (4.15) implies that each in can be scaled to define its own harmonic oscillator sub-albegra. The second condition implies, according to (4.13), that the annihilators and creators for and in the subspace commute amongst themselves:

Given such a decompostion a total Hilbert space for the field theory can be defined as the space of finite norm sums of possibly infinitely many states of the form

where is a state such that for all in The state as in

classical case, is called a Fock vacuum and Hilbert space, is called a Fock space. The representation of the field operator on this Fock space is hermitian and satisfies the canonical commutation relations in sense of Colombeau generalized function.

4.2 Defining distributional outgoing modesFor illustration we consider the non-rotating, uncharged -dimensional SAdS BH with a distributional line element

where

where is the

metric of the ( )-sphere, and the AdS curvature radius squared is related to the cosmological constant by The parameter is proportional to the

mass of the spacetime: where

The distributional Schwarzschild geometry corresponds to The corresponding equation of motion (4.2) for massless case are


151

1183

1184

1185

1186

1187

1188

1189

1190119111921193

1194

11951196

1197

1198119912001201

1202120312041205

1206

1207

1208

12091210121112121213

152153

The time-independence and the spherical symmetry of the metric imply the canonical decomposition

where denotes the -dimensional scalar spherical harmonics, satisfying

where is the Laplace-Beltrami operator. Substituting the decomposition (4.22) into Eq. (6) one gets a radial wave equation

We define now a tortoise distributional coordinate by the relation

By using a tortoise distributional coordinate the Eq. (4.24) can be written in the form of a Schrödinger equation with the potential

Note that the tortoise distributional coordinate becomes to infinite Colombeau

constant at the horizon, i.e. as but its behavior at infinity is

strongly dependent on the cosmological constant: for asymptotically-flat

spacetimes, and finite Colombeau constant for the SAdS geometry.


154

1214

12151216

1217

1218

1219

1220

1221

12221223

1224

1225

1226

1227

1228

12291230

1231

1232

1233

1234

1235

1236

1237

155156

4.2.1. Boundary conditions at the horizon of the distributional SAdS BH geometry

For most spacetimes of interest the potential as i.e.

and in this limit solutions to the wave equation (4.26) behave as

Note that classically nothing should leave the horizon and thus classically only ingoing modes (corresponding to a plus sign) should be present, i.e.

Note that for non-extremal spacetimes, the tortoise coordinate tends to

where Therefore near the horizon, outgoing modes behave as

where Now Eq. (4.30) shows that outgoing modes is Colombeau

generalized function of class

5. ENERGY-MOMENTUM TENSOR CALCULATION BY USING COLOMBEAU DISTRIBUTIONAL MODELS

We shall assume now any distributional spacetime which is conformally static in both the asymptotic past and future. We will be considered distributional spacetime which is conformally flat in the asymptotic past, i.e.

where , are smooth functions and

, are the components of an arbitrary distributional spatial

metric. Note that we use the same labels and for coordinates in the asymptotic past and future only for simplicity; they are obviously defined on non-intersecting


157

12381239

1240

1241

1242

1243

12441245

1246

1247

1248

1249

1250

1251

12521253

1254125512561257125812591260

1261

1262

1263126412651266

158159

regions of the spacetime.) In each of these asymptotic regions the distributional field

can be written as where satisfies

where is the flat Laplace operator, is the Laplace operator associated with

the spatial metric , and the effective potential is given by

with , the scalar curvature associated with the spatial

distributional metric .

We assume now this condition: (i) the massless ( ) field with arbitrary coupling in spacetimes which are asymptotically flat in the past and asymptotically static in the future, i.e. and , as those describing the formation of a static BH from matter initially scattered throughout space, and

(ii) the massless, conformally coupled field ( and ). With this assumptions for

the potential, two different sets of positive-norm distributional modes, and ,

can be naturally defined by the requirement that they are the solutions of Eq. (5.2) which satisfy the asymptotic conditions:

and

where , , , and are Colombeau solutions of

satisfying the normalization


160

12671268

1269

1270

12711272

1273

1274

12751276

1277127812791280

1281128212831284

1285

1286

1287

1288

1289

1290

1291

161162

on a Cauchy surface in the asymptotic future. Note that each can be chosen to be real without loss of generality. There are reasonable situations where the distributional modes , given in Eq. (5.5), together with distributional modes fail to form a complete set of distributional normal modes. This happens whenever the operator

in Eq. (5.2) happens to possess normalizable, i.e. satisfying Eq.

(5.7) eigenfunctions with negative eigenvalues, . In this case, additional

positive-norm modes with the asymptotic behavior

and their complex conjugates are necessary in order to expand an arbitrary Colombeau solution of Eq. (5.1). As a direct consequence, at least some of the in-modes

(typically those with low ) eventually undergo an exponential growth. This

asymptotic divergence is reflected on the unbounded increase of the vacuum fluctuations,

where is the eigenfunction of Eq. (5.6) associated with the lowest negative eigenvalue allowed, is some positive constant, and is a dimensionless constant (typically of order unity) whose exact value depends globally on the spacetime structure

(since it crucially depends on the projection of each on the mode whose

; also depends on the initial state, here assumed to be the vacuum ). As one would expect, these wild quantum fluctuations give an important contribution to the vacuum energy stored in the field. In fact, the expectation value of its distributional energy-momentum tensor, in the asymptotic future is found to be dominated by this exponential growth:


163

1292

12931294

12951296129712981299

1300

1301

13021303

13041305

1306

1307

130813091310

1311

13121313131413151316

1317

1318

164165

where is the derivative operator compatible with the distributional metric (so that

), is the associated distributional Ricci tensor so that

and we have omitted the subscript in and for simplicity. The Eqs. (5.10-5.12), together with Eq. (5.9), imply that on time scales determined by , the vacuum fluctuations of the field should overcome any other classical source of energy, therefore taking control over the evolution of the background geometry through the semiclassical Einstein equations (in which is included as a source term for the distributional Einstein tensor). We are then confronted with a startling situation where the quantum fluctuations of a field, whose energy is usually negligible in comparison with classical energy components, are forced by the distributional background spacetime to play a dominant role. We are still left with the task of showing that there exist indeed well-behaved distributional background spacetimes in which the operator

possesses negative eigenvalues condition on which depends Eq (5.9). Experience from quantum mechanics tells us that this typically occurs when gets sufficiently negative over a sufficiently large region. It is easy to see from Eq. (5.3) that, except for very special geometries (as the flat one), one can generally find appropriate values of which make as negative as would be necessary in order to guarantee the existence of negative eigenvalues. For distributional BH spacetime using Eq. (5.9) - Eq. (5.12) one obtains


166

1319

1320

1321132213231324132513261327132813291330133113321333133413351336133713381339

167168

Remark 5.1. Note that in spite of the unbounded growth at in Eq. (5.13) - Eq. (5.16),

is covariantly conserved: . In the static case

, for instance for distributional BH geometry, this implies that the total vacuum energy is kept constant, although it continuously flows from spatial regions where its density is negative to spatial regions where it is positive.


169

1340

1341

1342

1343

13441345134613471348

170171

Remark 5.2. Note that the singular behavior at appearing in Eq. (5.13) - Eq. (5.16) leads only to asymptotic divergences, i.e. all the quantities remain finite everywhere except horizon.

6. DISTRIBUTIONAL SADS BH SPACETIME-INDUCED VACUUM DOMINANCE

6.1. Adiabatic expansion of Green functions

Using equation of motion Eq. (5.2) one can obtain corresponding distributional generalization of the canonical Green functions equations. In particular for the distributional propagator

one obtains directly

Special interest attaches to the short distance behaviour of the Green functions, such as in the limit with a fixed We obtain now an adiabatic

expansion of Introducing Riemann normal coordinates for the point

with origin at the point we have expanding

where is the Minkowski metric tensor, and the coefficients are all evaluated at Defining now

and its Colombeau-Fourier transform by

where one can work in a sort of localized momentum space. Expanding

(6.2) in normal coordinates and converting to -space, can readily be solved by iteration to any adiabatic order. The result to adiabatic order four (i.e., four derivatives of the metric) is


172

134913501351

1352135313541355135613571358

1359

1360

1361

1362

1363

1364

1365

136613671368

1369

13701371

1372

1373

1374

1375

1376

1377137813791380

1381

173174

where

and

we are using the symbol to indicate that this is an asymptotic expansion. One ensures that Eq. (6.5) represents a time-ordered product by performing the integral along the appropriate contour in figure 3. This is equivalent to replacing by Similarly, the adiabatic expansions of other Green functions can be obtained by using the other contours in figure 3. Substituting Eq. (6.6) into Eq. (6.5) gives

where

and, to adiabatic order 4,

with all geometric quantities on the right-hand side of Eq. (6.10) evaluated at


175

1382

1383

1384

13851386138713881389

1390

1391

13921393

1394

1395

1396

1397

176177

Fig. 3. The contour in the complex plane to be used in the evaluation of the

integral giving . The cross indicates the pole at .

If one uses the canonical integral representation

in Eq. (6.8), then the integration may be interchanged with the integration, and performed explicitly to yield (dropping the )

The function which is one-half of the square of the proper distance between

and while the function has the following asymptotic adiabatic expansion

Using Eq. (6.4), equation (6.12) gives a representation of

where is the distributional Van Vleck determinant


178

1398

1399

1400

1401

1402

1403

1404

1405

14061407

14081409

1410

1411

1412

1413

1414

1415

1416

1417

179180

In the normal coordinates about that we are currently using, reduces to

The full asymptotic expansion of to all adiabatic orders

are

with the other being given by canonical recursion relations which enable their adiabatic expansions to be obtained. The expansions (6.13) and (6.16) are, however, only asymptotic approximations in the limit of large adiabatic parameter

If (6.16) is substituted into (6.14) the integral can be performed to give the adiabatic expansion of the Feynman propagator in coordinate space:

in which, strictly, a small imaginary part i should be subtracted from . Since we have not imposed global boundary conditions on the distributional Green function Colombeau solution of (6.2), the expansion (6.17) does not determine the particular vacuum state in (6.1). In particular, the " " in the expansion of only ensures that (6.17) represents the expectation value, in some set of states, of a time-ordered product of fields. Under some circumstances the use of " " in the exact representation (6.14) may give additional information concerning the global nature of the states.

6.2. Effective action for the quantum matter fields in curved distributional spacetime

As in classical case one can obtain Colombeau generalized quantity called the effective action for the quantum matter fields in curved distributional spcetime, which, when functionally differentiated, yields

To discover the structure of , let us return to first principles, recalling the Colombeau path-integral quantization procedure such as developed in []. Our notation will imply a treatment for the scalar field, but the formal manipulations are identical for fields of higher spins. Note that the generating functional


181

1418

14191420

1421

142214231424142514261427

1428

1429

143014311432

1433143414351436

1437

143814391440144114421443

1444

1445

1446144714481449

1450

1451

182183

was interpreted physically as the vacuum persistence amplitude The

presence of the external distributional current density can cause the initial vacuum

state to be unstable, i.e. it can bring about the production of particles. In flat

space, in the limit no particles are produced, and one have the normalization condition

However, when distributional spacetime is curved, we have seen that, in general,

even in the absence of source currents J. Hence (6.19) will no longer apply. Path-integral quantization still works in curved distributional spacetime; one simply treats in

(6.19) as the curved distributional spacetime matter action, and as a current density

(a scalar density in the case of scalar fields). One can thus set = 0 in (6.19) and examine

the variation of

Note that

From (6.22) and (6.23) one obtains directly

Noting that the matter action appears exponentiated in (6.19), one obtains directly

and


184

1452

14531454145514561457

1458

1459

1460

1461

1462

14631464146514661467

1468

1469

1470

1471

1472

1473

1474

1475

1476

1477

1478

1479

185186

Following canonical calculation one obtains

where the proportionality constant is metric-independent and can be ignored. Thus we obtain

In (6.28) is to be interpreted as an Colombeau generalized operator which acts on an

linear space of generalized vectors normalized by

in such a way that

Remark 6.1.Note that the trace of an Colombeau generalized operator which acts on a linear space is defined by

Writing now the Colombeau generalized operator as

by Eq. (6.14) one obtains

Now, assuming to have a small negative imaginary part, we obtain


187

1480

1481

1482

1483

1484

1485

14861487

1488

1489

1490

1491

1492

14931494

1495

1496

1497

1498

1499

1500

1501

1502

1503

188189

where is the exponential integral function.

Remark 6.2. Note that for

where is the Euler's constant. Substituting now (6.35) into (6.34) and letting we obtain

which is correct up to the addition of a metric-independent infinite large Colombeau constant that can be ignored in what follows. Thus, in the generalized De Witt-Schwinger

representation (6.33) or (6.14) we have

where the integral with respect to brings down the extra power of that appears in

Eq. (6.36). Returning now to the expression (6.28) for using Eq. (6.37) and Eq.(6.31) we get

Interchanging the order of integration and taking the limit one obtains

Colombeau quantity is called as the one-loop effective action. In the case of fermion effective actions, there would be a remaining trace over spinorial indices. From Eq. (6.39) we may define an effective Lagrangian density by

whence one get

6.3. Stress-tensor renormalization

Note that diverges at the lower end of the integral because the damping

factor in the exponent vanishes in the limit . (Convergence at the upper end is guaranteed by the that is implicitly added to in the De Witt-Schwinger

representation of In four dimensions, the potentially divergent terms in the DeWitt-

Schwinger expansion of are


190

1504

15051506

1507

15081509

1510

1511151215131514

1515

1516

151715181519

1520

1521

1522

1523152415251526152715281529

1530153115321533

1534

1535

191192

where the coefficients , and are given by Eq. (6.9) - Eq. (6.10). The remaining

terms in this asymptotic expansion, involving and higher, are finite in the limit

Let us determine now the precise form of the geometrical terms, to compare them with the conventional gravitational Lagrangian that appears in (2.38). This is a delicate matter because (6.48) is, of course, infinite. What we require is to display the divergent terms in the form geometrical object . This can be done in a variety of ways. For

example, in dimensions, the asymptotic (adiabatic) expansion of is

of which the first terms are divergent as If is treated as a variable which can be analytically continued throughout the complex plane, then we may take the limit

From Eq. (6.44) it follows we shall wish to retain the units of as (length) , even when It is therefore necessary to introduce an arbitrary mass scale and to rewrite Eq. (6.44) as

If the first three terms of Eq. (6.45) diverge because of poles in the - functions:

Denoting these first three terms by we have


193

1536

15371538

15391540154115421543

1544

1545

154615471548

1549

1550

155115521553

155415551556

1557

1558

1559

194195

The functions and are given by taking the coincidence limits of (6.9)-(6.10)

Finally one obtains

Special interest attaches to field theories in distributional spacetime in which the classical action is invariant under distributional conformal transformations, i.e.

From the definitions one has


and it is clear that if the classical action is invariant under the conformal transformations (6.50), then the classical stress-tensor is traceless. Because conformal transformations are essentially a rescaling of lengths at each spacetime point the presence of a mass and


196

1560

15611562

1563

1564

1565

1566

1567

156815691570

1571

1572

1573

1574

1575

1576

1577

1578

157915801581

197198

hence a fixed length scale in the theory will always break the conformal invariance. Therefore we are led to the massless limit of the regularization and renormalization procedures used in the previous section. Although all the higher order terms in the DeWitt-Schwinger expansion of the effective Lagrangian (6.45) are infrared divergent at

as we can still use this expansion to yield the ultraviolet divergent terms arising from and in the four-dimensional case. We may put immediately in the

and terms in the expansion, because they are of positive power for These terms therefore vanish. The only nonvanishing potentially ultraviolet divergent term is therefore

which must be handled carefully. Substituting for with from (6.48), and rearranging terms, we may write the divergent term in the effective action arising from (6.53) as follows

where

and

Finally one obtains

Note that from Eq. (3.42) for follows that

Thus for the case of the distributional Schwarzchild spesetime given by the distributional metric (3.40) using Eq. (6.57) and Eq. (6.58) for one obtains


199

1582158315841585158615871588158915901591

1592

159315941595

1596

1597

1598

1599

1600

1601

1602

1603

1604

1605

16061607

1608

160916101611

200201

This result is in a good agreement with Eq. (5.14) - Eq. (5.16).

7. COMCLUSIONS AND REMARKS

This paper dealing with an extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1]-[2], [5]-[7] and [14]-[15]. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically.

In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We have shown that a successful approach for dealing with curvature tensor valued distribution is to first impose admissible the nondegeneracy conditions on the metric tensor, and then take its derivatives in the sense of classical distributions in space.

The distributional meaning is then equivalent to the junction condition formalism. Afterwards, through appropriate limiting procedures, it is then possible to obtain well behaved distributional tensors with support on submanifolds of as we have shown for the energy-momentum tensors associated with the Schwarzschild spacetimes. The above procedure provides us with what is expected on physical grounds. However, it should be mentioned that the use of new supergeneralized functions (supergeneralized Colombeau algebras ) in order to obtain superdistributional curvatures, may renders a more rigorous setting for discussing situations like the ones considered in this paper.

The vacuum energy density of free scalar quantum field with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations has a singular behavior on BHs horizon :

We argue that this vacuum dominance may bear important astrophysical implications.

ACKNOWLEDGMENTSTo reviewers provided important clarifications.


202

1612

1613

1614

161516161617161816191620162116221623

1624162516261627

16281629163016311632163316341635

16361637163816391640164116421643164416451646

1647

1648

16491650

1651165216531654

1655

203204

REFERENCES

1. Colombeau JF. New Generalized Functions and Multiplication of Distributions. North Holland, Amsterdam: 1984.

2. Colombeau JF. Elementary Introduction to New Generalized Functions. North Holland, Amsterdam: 1985.

3. Tolman RC. Relativity, Thermodynamics and Cosmology. Clarendon Press, Oxford: 1962.

4. Landau LD, Lifschitz EM. The Classical Theory of Fields. Pergamon Press, Oxford: 1975.

5. Vickers JA, Wilson JP. A nonlinear theory of tensor distributions. gr-qc/9807068.6. Vickers JA, Wilson JP. Invariance of the distributional curvature of the cone under

smooth diffeomorphisms. Class. Quantum Grav. 1999; 16: 579-588.7. Vickers JA. Nonlinear generalized functions in general relativity. In: Grosser M,

Hörmann G, Kunzinger M, Oberguggenberger M., editors. Nonlinear Theory of Generalized Functions, Chapman & Hall/CRC Research Notes in Mathematics. 401, 275-290, eds. Chapman & Hall CRC, Boca Raton; 1999.

8. Geroch R, Traschen J. Strings and other distributional sources in general relativity. Phys. Rev. D 1987; 36: 1017-1031.

9. Balasin H, Nachbagauer H. On the distributional nature of the energy-momentum tensor of a black hole or what curves the Schwarzschild geometry? Class. Quant. Grav. 1993; 10: 2271-2278.

10. Balasin H, Nachbagauer H. Distributional energy-momentum tensor of the Kerr-Newman space-time family. Class. Quant. Grav. 1994; 11: 1453-1461.

11. Kawai T, Sakane E. Distributional energy-momentum densities of Schwarzschild space-time. Prog. Theor. Phys. 1997; 98: 69-86.

12. Pantoja N, Rago H. Energy-momentum tensor valued distributions for the Schwarzschild and Reissner-Nordstrøm geometries. Preprint gr-qc/9710072, 1997.

13. Pantoja N, Rago H. Distributional sources in General Relativity: Two point-like examples revisited. Preprint gr-qc/0009053, 2000.

14. Kunzinger M., Steinbauer R. Nonlinear distributional geometry. Acta Appl.Math. 2001.

15. Kunzinger M., Steinbauer R. Generalized pseudo-Riemannian Geometry. Preprint mathFA/0107057, 2001.

16. Grosser M, Farkas E, Kunzinger M., Steinbauer R. On the foundations of nonlinear generalized functions I, II. Mem. Am. Math. Soc. 2001; 153: 729.

17. Grosser M, Kunzinger M., Steinbauer R, Vickers J. A global theory of nonlinear generalized functions. Adv. Math. 2001; to appear.

18. Schwartz L. Sur l'impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 1954; 239: 847—848.

19. Gelfand IM, Schilov GE. Generalized functions. Vol. I: Properties and operations. Academic Press, New York, London: 1964.

20. Parker P. Distributional geometry. J. Math. Phys. 1979; 20: 1423--1426.21. Debney G, Kerr R, Schild A. Solutions of the Einstein and Einstein-Maxwell

equations. J. Math. Phys. 1969; 10: 1842-1854. 22. Heinzle JM, Steinbauer R. Remarks on the distributional Schwarzschild geometry.

Preprint gr-qc/0112047, 2001.23. Abramowitz M, Stegun IA, editors. Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables. 9th printing. Dover, New York: 1972.24. Bracewell R. Heaviside's Unit Step Function. In: The Fourier Transform and Its

Applications. 3rd ed. McGraw-Hill, New York: 1999, pp. 57-61.25. Israel W. Nuovo Cimento 1966; 44 1; Nuovo Cimento 1967; 48 463.


205

1656

1657165816591660166116621663166416651666166716681669167016711672167316741675167616771678167916801681168216831684168516861687168816891690169116921693169416951696169716981699170017011702170317041705170617071708

206207

26. Parker PE. J. Math. Phys. 1979; 20: 1423. 27. Raju CK. J. Phys. A: Math. Gen. 1982; 15: 1785.28. Geroch RP, Traschen J. Phys. Rev. D 1987; 38: 1017.29. Kawai T, Sakane E. Distributional Energy-Momentum Densities of Schwarzschild

Space-Time. Preprint grqc/9707029, 1998.30. Jacobson T. Introduction to Quantum Fields in Curved Spacetime and the Hawking

Effect. Available: http://arxiv.org/abs/0905.2975v2.31. Berti E, Cardoso V, Starinets AO. Quasinormal modes of black holes and black

branes. Available: http://arxiv.org/abs/gr-qc/0308048v3. 32. Lima WCC, Vanzella DAT. Gravity-induced vacuum dominance. Phys. Rev. Lett.

2010; 104: 161102. Available: http://arxiv.org/abs/1003.3421v1. 33. Choquet-Bruhat Y. General Relativity and the Einstein Equations. OUP Oxford: Dec

4, 2008 - Mathematics - 816 pages.34. Biswas T. Physical Interpretation of Coordinates for the Schwarzschild Metric.

Available: http://arxiv.org/abs/0809.1452v1.35. Hajicek P. An Introduction to the Relativistic Theory of Gravitation. Springer Science

& Business Media: Aug 26, 2008 - Mathematics - 280 pages.36. Müller T, Grave F. Catalogue of Spacetimes, Institut für Visualisierung und

Interaktive Systeme. Available: http://xxx.lanl.gov/abs/0904.4184v2, http://www.vis.uni-stuttgart.de/~muelleta/CoS/.


208

170917101711171217131714171517161717171817191720172117221723172417251726172717281729

17301731173217331734173517361737173817391740174117421743174417451746174717481749175017511752175317541755175617571758175917601761

209210

http://www.vis.uni-stuttgart.de/~muelleta/CoS/

http://xxx.lanl.gov/abs/0904.4184v2

http://arxiv.org/abs/0809.1452v1


http://arxiv.org/abs/gr-qc/0308048v3


APPENDIXExpressions for the Colombeau quantities and

in terms of and .

Let us introduce now Colombeau generalized metric which has the form

Th

e Colombeau scalars and in terms of

Colombeau generalized functions are expressed as

R A

2r 2A

A 3B

B

2r2AC D

2

AB A

A 2 B

B

12B

B

2 2A

B

AB 1

2A

A B

B

,

R R A2

212A

A 14

A

A 12

AB

AB 1r

A

A

2

2 A2

21r12

A

A 2B

B 1r2AC D

2

AB 12

AB

AB

12B

B 14

B

B

2

A2

212A

A 14

A

A 12

AB

AB B

B 12

B

B

2

12B

B 1r

A

A

2B

B

2

,

R R

A2

2A

A 12

A

A

2 2A

2

21rA

A 12

AB

AB

2

4A2

2

1rB

B 1r2AC D

2

AB 14

B

B

2 2

2 A2

21rA

A 2B

B

12

AB

AB B

B

12B

B

2 12

B

B

2

.

A. 2

Here


211

176217631764

1765

1766

1767

17681769

17701771

1772

212213

Assume that

From Eq. (A.2) - Eq. (A.4) one obtains


214

1773

1774

1775

1776

1777

1778

1779

1780

215216

2171781

218

Documents

Distributional SAdS BH spacetime-induced vacuum dominance 3ДОМRED1349769084-SDI Paper Template 2003